Hydrostatic Approximation Research Articles

On the role of the stratosphere in the process of overflow of mesoscale mountains

Abstract. A 2-D, two- and three-layer stratified airflow over a mountain of arbitrary shape is considered on the assumptions that upstream wind velocity and static stability within each layer are constant (Long's model). The stratosphere is simulated by an infinitely deep upper layer with enhanced static stability. The analytical solution for the stream function, as well as first (linear) and second order approximations to the wave drag, are obtained in hydrostatic limit N1L/U0→∞, where N1 is the Brunt-Väsälä frequency in the troposphere, L is a characteristic length of the obstacle, and U0 is upstream velocity. The results of numerical computations show the principal role of long waves in the process of interaction between the model layers for a typical mesoscale mountains for which the hydrostatic approximation proves valid in a wide range of flow parameters, in accordance with the earlier conclusions of Klemp and Lilly (1975). Partial reflection of wave energy from the tropopause produces strong influence on the value of wave drag for typical middle and upper tropospheric lapse rates, leading to a quasi-periodic dependance of wave drag on a reduced frequency ( is tropopause height) in the troposphere. The flow seems to be statically unstable for k≥2 for sufficiently large obstacles (whose height exceeds 1 km). In this case, vast regions of rotor motions and strong turbulence are predicted from model calculations in the middle troposphere and the lower stratosphere. The model calculations also point to a testify for possible important role of nonlinear effects associated with finite height of the mountain on the conditions of wave drag amplification in the process of overflow of real mountains.

Investigation of Transformations of the Local Regions of Pollution in the Northwest Part of the Black Sea Caused by the Motion of a Cyclone

By the method of mathematical simulation, we study the evolution of local discharges of pollutants on the northwest shelf of the Black Sea induced by the motion of a cyclone. We use the nonlinear equations of motion of a homogeneous viscous fluid in the hydrostatic approximation and the equation of turbulent diffusion. A cyclonic formation is represented by a moving axisymmetric area of low pressures. The discharges of pollutants are caused by the action of instantaneous sources located on the sea surface. We perform the analysis of dependences of the paths of motion of polluted regions, the periods of their dissipation, and the depths of penetration of pollutants on the bottom topography and the intensity of diffusion processes.

Regional circulation of the Monterey Bay region: Hydrostatic versus nonhydrostatic modeling

Ocean circulation off the west coast of the United States is driven by a variety of mechanisms, the most important of which are the seasonally varying local wind stress and coastal irregularities. Remote forcing is also important and expressed through the open boundary conditions. We use a nonhydrostatic, Dietrich/Center for Air‐Sea Technology (DieCAST) ocean model to simulate the regional circulation in the vicinity of Monterey Bay, California. Satellite images often show a cyclonic eddy in the bay and an anticyclonic eddy outside the bay during spring and summer. We compare the simulation results with observed mooring and HF radar‐derived velocity data. The coastal geometry plays an important role in the generation and movement of coastal eddies. Quantitative comparisons between hydrostatic and nonhydrostatic models are made to investigate the importance of the nonhydrostatic effects in coastal ocean simulation. The results show that both the Sur Ridge area and the Monterey Submarine Canyon contribute significantly to nonhydrostatic effects and small‐scale features. The strong nonhydrostatic and small‐scale features result from the California Undercurrent flows across sloping bathymetry and interactions with near‐surface California Current in summer. Rapid changes in slope in the presence of strong flows cause vertical acceleration, which violates the hydrostatic approximation. Surface‐trapped nonhydrostatic fronts also occur frequently in the shallow ocean during winter. These effects have seasonal variation and cannot be ignored in coastal ocean modeling with complex bathymetry.

Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi‐hydrostatic and non‐hydrostatic

AbstractWe study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones. The HPEs make spherical geopotential and shallow atmosphere approximations in addition to the hydrostatic approximation. As is well known, a consistent application of the shallow atmosphere approximation requires omission of those Coriolis terms that vary as the cosine of latitude and of certain other terms in the components of the momentum equation. An approximate model is here regarded as consistent if it formally preserves conservation principles for axial angular momentum, energy and potential vorticity, and (following R. Müller) if its momentum component equations have Lagrange's form. Within these criteria, four consistent approximate global models, including the HPEs themselves, are identified in a height‐coordinate framework. The four models, each of which includes the spherical geopotential approximation, correspond to whether the shallow atmosphere and hydrostatic (or quasi‐hydrostatic) approximations are individually made or not made. Restrictions on representing the spatial variation of apparent gravity occur. Solution methods and the situation in a pressure‐coordinate framework are discussed. © Crown copyright 2005.

The Vorticity–Velocity Gradient Cofactor Tensor and the Material Invariant of the Semigeostrophic Theory

Abstract A new derivation and interpretation of the semigeostrophic (SG) material invariant in the theory of geophysical flows is introduced. First, a generalized three-dimensional equation of the SG dynamics is established and the generalized equations for the rate of change of vorticity and for the rate of change of the velocity gradient cofactor tensor are obtained. Next, a conservation equation for the vorticity–velocity gradient cofactor tensor (denoted Ξ̃) is derived. The specific potential Ξ̃, that is, Ξ̃ in the reference configuration per unit of mass, is defined and an expression for its rate of change is obtained. The SG invariant is interpreted as the vertical component of the specific potential Ξ̃. Under the SG assumptions (advection of the geostrophic velocity, hydrostatic, and f-plane approximations) this vertical component is materially conserved in the SG flow. The generalized SG invariant (i.e., the specific potential Ξ̃) differs conceptually from the Beltrami–Rossby–Ertel specific potential vorticity. Its conservation in the SG flow seems to be highly dependent on the SG assumptions, especially on the f-plane approximation and on the horizontal nature of the geostrophic velocity.

Response of the southwestern Japan/East Sea to atmospheric pressure

Abstract The response of the southwestern Japan/East Sea (JES) to atmospheric pressure ( P atm ) and wind-stress ( τ ⇒ ) forcing is investigated by analyzing 2-year bottom pressure ( P bot ) data and coastal tide-gauge records. Coherence analyses between P bot data reveal that the response of the southwestern JES is nearly uniform at frequencies lower than 0.6 cycles per day (cpd). The Ulleung Basin (UB) average P bot ( P ¯ bot ) departs significantly from inverted-barometer (IB) response to the basin average P atm ( P ¯ atm ) at frequency bands from 0.2 to 0.7 cpd. The coherence between P ¯ atm and P ¯ bot has maximum value at 0.2 cpd. Multiple coherence analysis, applied with P ¯ atm and UB average τ ⇒ ( τ ⇒ ¯ ) as inputs and P ¯ bot as output, reveals that P ¯ atm is the most significant forcing, with a peak at frequencies between 0.2 and 0.3 cpd. A simple model [Garrett, 1983. Variable sea level and strait flows in the Mediterranean: a theoretical study of the response to meteorological forcing. Oceanologica Acta 6, 79–87; Lyu et al., 2002. Atmospheric pressure-forced subinertial variations in the transport through the Korea Strait. Geophysical Research Letters 29, 10.1029/2001GL014366] is used to investigate the limiting role of the three straits on the JES response to P atm . Coastal sea-level ( η ) data inside the JES as well as outside the straits demonstrate that the JES responds with a damped Helmholtz-like resonance. The resonance frequency predicted by this simple model is near the frequency of maximum coherence between P ¯ atm and η ¯ d , estimated from P ¯ bot by the hydrostatic approximation. Phase relations and response function gain between these variables confirm the applicability of this simple model to the JES for low-frequency bands below the Helmholtz-like resonance frequency. At higher frequencies, the response relaxes back toward IB, which suggests the mass field adjusts internally within the JES.

Simulation of the Evolution of Wave Fields Induced by a Moving Cyclone over the Northwest Shelf of the Black Sea

The wave fields generated by cyclonic atmospheric disturbances in the region of the northwest shelf of the Black Sea are studied by using a σ-coordinate model. The problem is solved in the hydrostatic approximation with regard for nonlinearity and turbulent viscosity. To improve the space resolution of the model, we realize the procedure of embedded grids. It is shown that the action of a cyclone results in the formation of domains of pronounced surges whose shapes and location reflect the changes in the wind situation. The vertical structure of wave currents first depends only on the location of the atmospheric disturbance over the water area but, in the course time, begins to reflect specific features of the bottom topography.

Comment on “Variations of total solar irradiance produced by structural changes in the solar interior”

In a recent Eos article, Sofia [2004] argues for the influence on irradiance variation of global changes in the Sun's structure associated with its magnetic dynamo. These changes would act in addition to the relatively well understood modulation by dark sunspots and bright faculae at the surface. His assessment of the present observational evidence for such a global change agrees with our earlier conclusion that it is not widely convincing at the present time [Foukal, 2003]. But Sofia's article also claims (1) that the numerical results obtained by him and his collaborators at Yale disagree with and correct earlier work, and (2) that a hydrostatic approximation is not adequate for variations on the 11‐year solar cycle timescale. These surprising claims are based on the results of recently published hydrostatic models [e.g., Sofia and Li, 2004] using the same mixing length approximation for convective heat transport used in earlier work [Spruit, 1982, 1991; Gilliland, 1988].

Sediment distribution study in the Gulf of Kachchh, India, from 3D hydrodynamic model simulation and satellite data

A fully 3D finite difference baroclinic hydrodynamic model ‘COSMOS’ is applied to the Gulf of Kachchh (GoK) to study tidal variation, ocean currents, residual tidal current, sea surface temperature distribution, etc. The ‘COSMOS’ is based on the hydrostatic and Boussinesq approximations and uses a vertical double sigma coordinate with a step-like grid. In addition to the momentum and continuity equations, the model solves two-transport equations for salinity and temperature and an equation of state to include baroclinic effects. The other objectives are quantitative assessment of suspended sediments by analyzing SeaWiFS data and understand the relation between residual tidal currents and erosion, movement and deposition of sediments. The model results showed that the dominant current system within GoK is controlled by the simulated tidal and residual tidal currents variation. The residual tidal current velocity reduces from the surface towards the bottom. The residual surface current velocity distribution displays cyclonic eddy continuity regions within the GoK. It is concluded that bottom topography variation plays an important role in determining the distribution of residual current velocity and thereby sediment distribution. The residual current velocity distribution map showed conformity with the gross geomorphic subdivisions and the sediment distribution map of Gulf of Kachchh. The suspended sediment plume pattern followed the current pattern obtained from modeling. The model results of the sea surface temperature of the December month showed good agreement with the temperature structure and the pattern obtained form NOAA/AVHRR data.

Mechanisms of Up-Valley Winds

Abstract The basic physical mechanisms governing the daytime evolution of up-valley winds in mountain valleys are investigated using a series of numerical simulations of thermally driven flow over idealized three-dimensional topography. The three-dimensional topography used in this study is composed of two, two-dimensional topographies: one a slope connecting a plain with a plateau and the other a valley with a horizontal floor. The present two-dimensional simulations of the valley flow agree with results of previous investigations in that the heated sidewalls produce upslope flows that require a compensating subsidence in the valley core bringing down potentially warmer air from the stable free atmosphere. In the context of the three-dimensional valley–plain simulations, the authors find that this subsidence heating in the valley core is the main contributor to the valley– plain temperature contrast, which, under the hydrostatic approximation, is the main contributor to the valley– plain pressure difference that drives the up-valley wind.

REGULARIZATION BY MONOTONE PERTURBATIONS OF THE HYDROSTATIC APPROXIMATION OF NAVIER–STOKES EQUATIONS

Due to the lack of regularity of the solutions to the hydrostatic approximation of Navier–Stokes equations, an energy identity cannot be deduced. By including certain nonlinear perturbations to the hydrostatic approximation equations, the solutions to the perturbed problem are smooth enough so that they satisfy the corresponding energy identity. The perturbations considered in this paper are of the monotone class. Three kinds of problems are then studied. To do that, we introduce a functional setting and show in every case that the set of smooth functions with compact support is dense in the space where we search for solutions. When the perturbations are small enough in a certain sense, the solutions of the perturbed problem are close to those of the original one. As a result, this gives a new proof of the existence of solutions to the hydrostatic approximation of Navier–Stokes equations. Finally, this regularization technique has been applied to the analysis of a one-equation hydrostatic turbulence model.

The Effect of Wind Shear and Curvature on the Gravity Wave Drag Produced by a Ridge

Abstract The analytical model proposed by Teixeira, Miranda, and Valente is modified to calculate the gravity wave drag exerted by a stratified flow over a 2D mountain ridge. The drag is found to be more strongly affected by the vertical variation of the background velocity than for an axisymmetric mountain. In the hydrostatic approximation, the corrections to the drag due to this effect do not depend on the detailed shape of the ridge as long as this is exactly 2D. Besides the drag, all the perturbed quantities of the flow at the surface, including the pressure, may be calculated analytically.

On distributions independent of xNxN in certain non-cylindrical domains and a de Rham lemma with a non-local constraint

We study distributions not depending on the x N -variable in an open set Ω ⊂ R N . It is assumed that Ω may be described through a very general function D : ω ↦ R , where ω ⊂ R N - 1 is any open set. We give a representation theorem for this kind of distributions and show how they are related to distributions defined in ω . A direct application of this theorem is the derivation of a de Rham-like lemma with a non-local constraint. These results can be applied to the analysis of hydrostatic approximation of Navier–Stokes equations.

On distributions independent of [formula omitted] in certain non-cylindrical domains and a de Rham lemma with a non-local constraint

We study distributions not depending on the x N -variable in an open set Ω ⊂ R N . It is assumed that Ω may be described through a very general function D : ω ↦ R , where ω ⊂ R N - 1 is any open set. We give a representation theorem for this kind of distributions and show how they are related to distributions defined in ω . A direct application of this theorem is the derivation of a de Rham-like lemma with a non-local constraint. These results can be applied to the analysis of hydrostatic approximation of Navier–Stokes equations.

Hydrostatic and non-hydrostatic studies of gravitational adjustment over a slope

In many numerical ocean models, the hydrostatic approximation is made. This approximation causes a considerable saving in computing time. However, for phenomena involving large vertical speeds, for many small scale phenomena, and in areas with weak stratification, the approximation becomes questionable. In this report, a σ -coordinate hydrostatic C-grid model is extended to include non-hydrostatic dynamics. The test cases involve gravitational adjustment of a downslope flow. The first test case has a simplified slope profile and no ambient stratification in the deep basin. The second test case has ambient stratification and more realistic topography. The differences between hydrostatic and non-hydrostatic simulations are described and discussed. It is shown that the shapes of the head and the body of density driven plumes are better preserved in the non-hydrostatic experiments. The wave propagation away from the plume head is considerably reduced when including non-hydrostatic effects.

An Analytical Model of Mountain Wave Drag for Wind Profiles withShear and Curvature

Abstract An analytical model is developed to predict the surface drag exerted by internal gravity waves on an isolated axisymmetric mountain over which there is a stratified flow with a velocity profile that varies relatively slowly with height. The model is linear with respect to the perturbations induced by the mountain, and solves the Taylor–Goldstein equation with variable coefficients using a Wentzel–Kramers–Brillouin (WKB) approximation, formally valid for high Richardson numbers, Ri. The WKB solution is extended to a higher order than in previous studies, enabling a rigorous treatment of the effects of shear and curvature of the wind profile on the surface drag. In the hydrostatic approximation, closed formulas for the drag are derived for generic wind profiles, where the relative magnitude of the corrections to the leading-order drag (valid for a constant wind profile) does not depend on the detailed shape of the orography. The drag is found to vary proportionally to Ri−1, decreasing as Ri decreas...

Sea Level Variations in the Western Mediterranean Studied by a Numerical Tidal Model of the Strait of Gibraltar

Aspects of the sea level changes in the western Mediterranean Sea are investigated using a numerical tidal model of the Strait of Gibraltar. As a prerequisite, the performance of this model, that is, a two-dimensional, nonlinear, two-layer, boundary-fitted coordinate numerical model based on the hydrostatic approximation on an f plane, is assessed in the simulation of mean and tidal circulation of the Strait of Gibraltar. The model is forced by imposing mean interface and surface displacements as well as M2, S2, O1, and K1 tidal components along the Atlantic and Mediterranean model open boundaries. Model results are compared with observations and with results obtained from a tidal inverse model for the eastern entrance of the Strait of Gibraltar. In general, good agreement is found. A sensitivity study performed by varying different model parameters shows that the model behaves reasonably well in the simulation of the averaged circulation. The model is then used to investigate the climatological sensitivity of the simulated dynamics in the Strait of Gibraltar to changes in the density difference between Atlantic and Mediterranean waters. For this purpose, given a certain density difference between Atlantic and Mediterranean waters, the authors iteratively searched for that sea level drop between the Atlantic and the Mediterranean that fulfills the mass balance of the Mediterranean. It is found that an increase of the density difference leads to an increase of the exchange flow and to an increase of the sea level drop between the two basins. A trend in the sea level drop of O(1 cm yr21), such as the one observed between 1994 and 1997, is explained by the model as the result of a trend of O(10 24 yr21) in the relative density difference between the Mediterranean and Atlantic waters. The observed north‐south asymmetry in this trend is also captured by the model, and it is found to arise from changes in the along-strait velocity. Results suggest that the dynamics within the Strait of Gibraltar cannot be neglected when sea level changes in the western Mediterranean basin are investigated.

How Sensitive Are Coarse General Circulation Models to Fundamental Approximations in the Equations of Motion?

The advent of high-precision gravity missions presents the opportunity to accurately measure variations in the distribution of mass in the ocean. Such a data source will prove valuable in state estimation and constraining general circulation models (GCMs) in general. However, conventional GCMs make the Boussinesq approximations, a consequence of which is that mass is not conserved. By use of the height‐pressure coordinate isomorphism implemented in the Massachusetts Institute of Technology general circulation model (MITGCM), the impact of non-Boussinesq effects can be evaluated. Although implementing a non-Boussinesq model in pressure coordinates is relatively straightforward, making a direct comparison between height and pressure coordinate (i.e., Boussinesq and non-Boussinesq) models is not simple. However, a careful comparison of the height coordinate and the pressure coordinate solutions ensures that only non-Boussinesq effects can be responsible for the observed differences. As a yardstick, these differences are also compared with those between the Boussinesq hydrostatic and models in which the hydrostatic approximation has been relaxed, another approximation commonly made in GCMs. Model errors (differences) caused by the Boussinesq and hydrostatic approximations are demonstrated to be of comparable magnitude. Differences induced by small changes in subgrid-scale parameterizations are at least as large. Therefore, non-Boussinesq and nonhydrostatic effects are most likely negligible with respect to other model uncertainties. However, because there is no additional cost incurred in using a pressure coordinate model, it is argued that non-Boussinesq modeling is preferable simply for tidiness. It is also concluded that even coarse-resolution GCMs can be sensitive to small perturbations in the dynamical equations.

Modelling solitons under the hydrostatic and Boussinesq approximations

AbstractAn examination of solitary waves in 3D, time‐dependant hydrostatic and Boussinesq numerical models is presented. It is shown that waves in these models will deform and that only the acceleration term in the vertical momentum equation need be included to correct the wave propagation. Modelling of solitary waves propagating near the surface of a small to medium body of water, such as a lake, are used to illustrate the results. The results are also compared with experiments performed by other authors. Then as an improvement, an alternative numerical scheme is used which includes only the vertical acceleration term. Effects of horizontal and vertical diffusion on soliton wave structure is also discussed. Copyright © 2003 John Wiley & Sons, Ltd.

Review of numerical methods for nonhydrostatic weather prediction models

¶Currently available computer power allows to run operational numerical weather prediction models at resolutions higher than 10 km. The aim of such high resolution modeling is the prediction of local weather, including orographically induced winds and local precipitation patterns. In this range the hydrostatic approximation is no longer valid and nonhydrostatic models have to be used instead. For several decades these models have been developed for research purposes only, but operational application is now reality. In this paper, the numerical methods used in current nonhydrostatic forecast models will be reviewed and some promising techniques in this field will be discussed. Special attention is given to aspects such as the choice of the vertical coordinate, the efficiency of algebraic solvers for semi-implicit time discretizations, and accurate and non-oscillatory advection schemes.