On the Discrete Normal Modes of Quasigeostrophic Theory

The discrete baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes topography into account.

Poleward Propagation of Typhoon-Induced Near-Inertial Waves in the Northern South China Sea

One of the main responses of the ocean to typhoons is the generation of near-inertial waves (NIWs), whose intrinsic frequency is close to the local inertial frequency. Based on the mooring observations, we carefully investigated the spatial–temporal variations in NIWs in the northern South China Sea (SCS) after Typhoon “Haima,” which passed through the northern SCS from October 20 to 21, 2016, with its track parallel to the mooring array on its northeast side. Moorings in different locations responded differently to this typhoon, with stronger NIWs found on the continental slope to the shelf break and relatively weak NIWs found on the shallow continental shelf. Strong NIWs appeared twice within a short period and showed different characteristics and frequencies. The first NIWs were generated locally with blue-shifted (super-inertial) frequencies. These initial NIWs were dominated by the first three baroclinic modes and rapidly weakened due to their propagation away and local dissipation. However, the second NIWs mainly consisted of higher baroclinic modes with red-shifted (sub-inertial) frequencies. The analysis of the mean background flow revealed that these red-shifted NIWs were excited at low latitudes, and subsequently advected by the poleward background flow to moorings whose latitudes were higher than their critical latitudes. Higher-mode (n > 3) NIWs were more easily carried away by mean background flow, and, at the same time, high-mode NIWs propagated downward to the subsurface layer, leading to the significant elevation of velocity shear therein. Given these findings, the mean background flow may provide an important route to redistributing the energy of the upper ocean imported from typhoons.

A nonlinear multi-scale theory of atmospheric blocking: Structure and evolution of blocking linked to meridional and vertical structures of storm tracks

This paper examines the impact of the meridional and vertical structures of a preexisting upstream storm track (PUST) organized by preexisting synoptic-scale eddies on eddy-driven blocking in a nonlinear multi-scale interaction model. In this model, the blocking is assumed, based on observations, to be comprised of barotropic and first baroclinic modes, whereas the PUST consists of barotropic, first baroclinic and second baroclinic modes. It is found that the nonlinearity (dispersion) of blocking is intensified (weakened) with increasing amplitude of the first baroclinic mode of the blocking itself. The blocking tends to be long-lived in this case. The lifetime and strength of blocking are significantly influenced by the amplitude of the first baroclinic mode of blocking for given basic westerly winds (BWWs), whereas its spatial pattern and evolution are also affected by the meridional and vertical structures of the PUST.It is shown that the blocking mainly results from the transient eddy forcing induced by the barotropic and first baroclinic modes of PUST, whereas its second baroclinic mode contributes little to the transient eddy forcing. When the PUST shifts northward, eddy-driven blocking shows an asymmetric dipole structure with a strong anticyclone/weak cyclone in a uniform BWW, which induces northward-intensified westerly jet and storm track anomalies mainly on the north side of blocking. However, when the PUST has no meridional shift and is mainly located in the upper troposphere, a north-south anti-symmetric dipole blocking and an intensified split jet with maximum amplitude in the upper troposphere form easily for vertically varying BWWs without meridional shear.

Spatiotemporal structure of Baltic free sea level oscillations in barotropic and baroclinic conditions from hydrodynamic modelling

Free sea level oscillations in barotropic and baroclinic conditions were examined using numerical experiments based on a 3-D hydrodynamic model of the Baltic Sea. In a barotropic environment, the highest amplitudes of free sea level oscillations are observed in the northern Gulf of Bothnia, eastern Gulf of Finland, and south-western Baltic Sea. In these areas, the maximum variance appears within the frequency range corresponding to periods of 13–44 h. In a stratified environment, after the cessation of meteorological forcing, water masses relax to the equilibrium state in the form of mesoscale oscillations at the same frequencies as well as in the form of rapidly decaying low-frequency (seasonal) oscillations. The total amplitudes of free baroclinic perturbations are significantly larger than those of barotropic perturbations, reaching 15–17 cm. Contrary to barotropic, oscillations in baroclinic conditions are strongly pronounced in the deep-water areas of the Baltic Sea proper. Specific spatial patterns of amplitudes and phases of free barotropic and baroclinic sea level oscillations identified them as progressive–standing waves representing barotropic or baroclinic modes of gravity waves and topographic Rossby waves.

Does the interior of the ocean hide a major part of the eddy field?

<p>In ocean research, mesoscale eddies typically are detected through surface signatures based on satellite data. The assumption is that most eddies are surface intensified and have a vertical structure consistent with a surface intensified mode. However, in-situ eddy observations, especially in the tropical oceans, showed that the vertical eddy structure is often more complex than previously assumed (higher baroclinic modes), and a diverse subsurface eddy field is present, which does not show any surface signatures at all. Our objective here is a first step towards a quantification of the occurrence of subsurface relative to surface eddies. To do this, we use an actively eddying model to compare the subsurface eddy field to its surface signatures in order to be able to estimate which vertical eddy structures prevail and how much of the eddy field is hidden in the subsurface. In addition, the model results are compared against an unprecedented assemblage of observations of subsurface eddies in the tropical oceans. In a first step we focus on eddies in the model that are detectable at the surface for more than 120 days. We found that around 60 % of the detected eddies have a vertical structure associated with a surface intensified mode as previously assumed which are characterized by a strong surface signature. Around 40 % of the eddy field have a vertical structure associated to a higher baroclinic mode. They are often called “intrathermocline” eddies and are characterized by a rather weak surface signature. In a second step we track subsurface eddies (lifetime > 120 days) in the model by identifying density layer thickness anomalies and connect them with possible surface signatures. Around 30 % of the total eddy field of the model, are hidden in the subsurface with no detectable surface signature. In conclusion, our results show that subsurface eddies form a substantial contribution to the total eddy field. Consequently it is difficult to estimate the impact of the eddy field on the ocean when only working with surface based satellite data.</p>

The Energy Budget of an Altimeter-Derived Baroclinic Tide Model

<p>Over the last few years a number of groups have created maps of the baroclinic tide from satellite altimeter measurements of sea-surface height (SSH). These maps can be used as predictive models for the baroclinic tides, e.g., for removing aliased tidal signals from altimetry, but they can also be used to diagnose aspects of the tidal dynamics. This presentation uses the High Resolution Emprical Tide (HRET) model to compute the phase speed, energy, energy flux, and energy flux divergence of the first few baroclinic modes for the M2, S2, K1, and O1 tides, and compares these with independent estimates from the literature.</p><p>The phase speed of the waves in HRET are compared with the theoretically-predicted phase speeds computed from stratification. For the mode-1 M2 waves which are determined most accurately, the theoretical and observed phase speeds agree very well; however, there is a small bias, namely, the theoretical phase speed exceeds the observed phase speed by 1 to 2%. This offset could reflect either a methodological estimation bias, issues with the data used to compute the theoretical phase speed, or a limitation of the theory for the vertical modes.</p><p>The phase speed results provide some confidence in the usefulness of linear wave dynamics for interpreting the HRET SSH. Using a simplified form of the momentum equations, the area-integrated kinetic plus potential energy of the mode-1 M2 tide is found to be 43 PJ, larger than in other baroclinic tide models, and with nearly isotropic directional distribution. For mode 1, the divergence of the energy flux diagnosed from HRET agrees well with previous estimates based on the barotropic tides. For the most accurately-determined mode-1 M2 tide, the results provide new information about sources and sinks of baroclinic energy along the continental shelves, and they are used to examine the accuracy of a commonly-used approximation of the baroclinic energy flux.</p>

Discrete Normal Mode Decompositions in Quasigeostrophic Theory

<p>The baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple equivalent barotropic modes, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state. Using this complete basis, we are able to obtain a series expansion to the potential vorticity of Bretherton (with Dirac delta contributions). We compare the convergence and differentiability properties of these complete modes with various other modes in the literature. We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. In the process, we introduce the concept of the quasigeostrophic phase space which we define to be the space of all possible quasigeostrophic states. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes prescribed boundary buoyancy gradients into account.</p>

Spatio-temporal structure of Baltic free sea level oscillations in barotropic and baroclinic conditions from hydrodynamic modelling

Free sea level oscillations in barotropic and baroclinic conditions were examined using numerical experiments based on a 3D hydrodynamic model of the Baltic Sea. In a barotropic environment, the highest amplitudes of free sea level oscillations are observed in the northern Gulf of Bothnia, eastern Gulf of Finland, and south-western Baltic Sea. In these areas, the maximum variance appears within the frequency range corresponding to periods of 13–44 hr. In a stratified environment, after the cessation of meteorological forcing, water masses relax to the equilibrium state in the form of mesoscale oscillations at the same frequencies as well as in the form of rapidly decaying low-frequency (seasonal) oscillations. The total amplitudes of free baroclinic perturbations are significantly larger than those of barotropic perturbations, reaching 15–17 cm. Contrary to barotropic, oscillations in baroclinic conditions are strongly pronounced in the deep-water areas of the Baltic Sea Proper. Specific spatial patterns of amplitudes and phases of free barotropic and baroclinic sea level oscillations identified them as progressive-standing waves representing barotropic or baroclinic modes of gravity waves and topographic Rossby waves.

Determination of Spatiotemporal Variability of the Indian Equatorial Intermediate Current

Using 4-yr mooring observations and ocean circulation model experiments, this study characterizes the spatial and temporal variability of the Equatorial Intermediate Current (EIC; 200–1200 m) in the Indian Ocean and investigates the causes. The EIC is dominated by seasonal and intraseasonal variability, with interannual variability being weak. The seasonal component dominates the midbasin with a predominant semiannual period of ~166 days but weakens toward east and west where the EIC generally exhibits large intraseasonal variations. The resonant second and fourth baroclinic modes at the semiannual period make the largest contribution to the EIC, determining the overall EIC structures. The higher baroclinic modes, however, modify the EIC’s vertical structures, forming multiple cores during some time periods. The EIC intensity has an abrupt change near 73°E, which is strong to the east and weak to the west. Model simulation suggests that the abrupt change is caused primarily by the Maldives, which block the propagation of equatorial waves. The Maldives impede the equatorial Rossby waves, reducing the EIC’s standard deviation associated with reflected Rossby waves by ~48% and directly forced waves by 20%. Mode decomposition further demonstrates that the semiannual resonance amplitude of the second baroclinic mode reduces by 39% because of the Maldives. However, resonance amplitude of the four baroclinic mode is less affected, because the Maldives fall in the node region of mode 4’s resonance. The research reveals the spatiotemporal variability of the poorly understood EIC, contributing to our understanding of equatorial wave–current dynamics.

Baroclinic Modes over Rough Bathymetry and the Surface Deformation Radius

The deformation radius is widely used as an indication of the eddy length scale at different latitudes. The radius is usually calculated assuming a flat ocean bottom. However, bathymetry alters the baroclinic modes and hence their deformation radii. In a linear quasigeostrophic two-layer model with realistic parameters, the deep flow for a 100-km wave approaches zero with a bottom ridge roughly 10 m high, leaving a baroclinic mode that is mostly surface trapped. This is in line with published current meter studies showing a primary EOF that is surface intensified and has nearly zero flow at the bottom. The deformation radius associated with this “surface mode” is significantly larger than that of the flat bottom baroclinic mode. Using World Ocean Atlas data, the surface radius is found to be 20%–50% larger over much of the globe, and 100% larger in some regions. This in turn alters the long Rossby wave speed, which is shown to be 1.5–2 times faster than over a flat bottom. In addition, the larger deformation radius is easier to resolve in ocean models.