We suggest that, in situations where the wind is blowing from land to sea, the wind experiences some sort of inertial oscillations. This is shown by using a simplified version of the equations in the atmospheric boundary layer. In order to confirm that this effect can be present in a typical climate and weather prediction model, we have used the one-column version of the ECMWF global model. The

model was used in a “Lagrangian” framework, being advected at constant speed from land to sea. Inertial oscillations, in the wind at 10 metres above the surface, are produced by this simulation. This confirms the initial conjecture. Considering the mean wind speed, the time required by the surface boundary layer to adapt to the new (from land to sea) situation is of the order of a few hours

Cavaleri,, L.

Teixeira, J.

English

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IL NUOVO CIMENTO Vol. 25 C, N. 2 Marzo-Aprile 2002Oscillations in an offshore blowing wind(∗)L. Cavaleri(1)(∗∗) and J. Teixeira(2)(∗∗∗)(1) Istituto Studio Dinamica Grandi Masse - S. Polo 1364, 30125 Venezia, Italy(2) Naval Research Laboratory - Monterey, CA 93943-5502, USA(ricevuto il 15 Maggio 2000; revisionato il 27 Luglio 2001; approvato il 15 Novembre 2001)Summary. — We suggest that, in situations where the wind is blowing from landto sea, the wind experiences some sort of inertial oscillations. This is shown by usinga simplified version of the equations in the atmospheric boundary layer. In order toconfirm that this effect can be present in a typical climate and weather predictionmodel, we have used the one-column version of the ECMWF global model. Themodel was used in a “Lagrangian” framework, being advected at constant speedfrom land to sea. Inertial oscillations, in the wind at 10 metres above the surface,are produced by this simulation. This confirms the initial conjecture. Consideringthe mean wind speed, the time required by the surface boundary layer to adapt tothe new (from land to sea) situation is of the order of a few hours.PACS 92.60.Gn – Winds and their effects.PACS 92.10.Kp – Sea-air energy exchange processes.PACS 92.10.Sx – Coastal and estuarine oceanography.1. – IntroductionThe WAM wave model [1] is operational at the European Centre for Medium-RangeWeather Forecasts (ECMWF, Reading, UK) both at the global and European scale,the latter one providing a detailed description of the wave fields in the various localsub-basins. The quality of the results shows a marked variability. While the resultsare pretty good in the oceans, and in general in the large open spaces [2a], in the moreenclosed basins we find a permanent underestimation of the wave conditions (wave heightsand periods). In particular in the Mediterranean Sea the model wave heights are lowerthan the ones derived from satellite altimetry by about 30% [2b]. Looking more indetail to the distribution, the error seems to decrease with fetch. Given the physical(∗) The authors of this paper have agreed to not receive the proofs for correction.(∗∗) E-mail: L.Cavaleri@isdgm.ve.cnr.it(∗∗∗) Formerly at European Centre for Medium-Range Weather Forecasts, ReadingBerksh., UK.c© Societa` Italiana di Fisica 175176 L. CAVALERI and J. TEIXEIRAconnection between wind and waves, this suggests the possibility that the model speedof an offshore blowing wind is underestimated soon off the coast, the error decreasingwhile air propagates over the sea. It was suggested that this could be due to the timerequired by the model boundary layer to adapt to the new situation after the transitionfrom land to sea. We have explored this hypothesis using the one-column version ofECMWF. Apart from providing a reply to the original question, this has allowed adetailed analysis of the transient situation. In particular we found, first from an analysisof the equations, then from the numerical experiments, some results not reported in theliterature at our disposal, and that we suggest as a possibility to be verified.In what follows sect. 2 is devoted to a compact description of the aspects of the modelrelevant for our purposes. In sect. 3 we describe the organisation of the tests, while insect. 4 we report our main findings. Sections 4 and 5 include a discussion of the results.2. – The one-column modelThe one-column version of the ECMWF meteorological model, henceforth referredto as 1C ([3]; see also [4], for an overall description of the global model), has proved tobe a very useful tool for testing and experimenting with the physical parametrizationand with new developments in the vertical discretisation of the equations. Examples ofsuccessful applications in boundary layers studies are given by Duynkerke et al. [5] andTeixeira [6], in Lagrangian simulations by Bretherton et al. [7].2.1. The model equations. – The 1C model integrates the following set of equationsfor a column of the atmosphere:∂u∂t+ η˙∂u∂η= f(v − Vg) + Pu + Fu ,(1)∂v∂t+ η˙∂v∂η= −f(u− Ug) + Pv + Fv ,(2)∂T∂t+ η˙∂T∂η=RCpTωp+ PT + FT ,(3)∂q∂t+ η˙∂q∂η= Pq + Fq ,(4)where η is the hybrid vertical coordinate (see [8]), u and v are the zonal and meridionalcomponents of wind, T is the temperature, q is the specific humidity, ω is the verticalvelocity in pressure coordinates, p is the pressure, R is the gas constant (taking intoaccount the effect of moisture, liquid water and ice), Ug and Vg are the zonal and merid-ional components of geostrophic wind, Pi is the tendency due to physics, Fi is is theadditional forcing term, in this case the horizontal advection of the mean variables. f isthe Coriolis term defined as f = 2Ω sin(Φ), where Ω is the Earth’rotation rate and Φ isthe latitude. The geostrophic wind is defined asUg = − 1ρf∂p∂y,(5)OSCILLATIONS IN AN OFFSHORE BLOWING WIND 177Vg =1ρf∂p∂x(6)and in practice represents the pressure gradient term. The one-column model has thefollowing set of physical parametrizations: the radiation parametrization is based on aband emissivity scheme for the long-wave parametrization and a delta Eddington schemefor the short-wave parametrization [9]; the cloud scheme is the one described in detail inTiedkte [10]. The moist convection is parametrized by using a mass-flux scheme and isdescribed in Tiedkte [11]; the vertical diffusion parametrization is based on a K-diffusionapproach [12, 13] and on a Monin-Obukhov–type closure for the surface layer [14]; thesub-grid scale orographic scheme is described in Lott and Miller [15] and the soil/surfacescheme is described in detail in Viterbo and Beljaars [16] and Viterbo et al. [17].2.2. Some general considerations. – Before proceeding to the numerical integration ofthe above set of equations, it is convenient to analyse their structure for a qualitativeanticipation of the results.Inertial oscillations are common in situations where the dominant terms in the mo-mentum equations are the tendency, the Coriolis term and the pressure gradient term.Consider the following simplified version of the previous momentum equations:dudt= fv ,(7)dvdt= f(Ug − u) ,(8)where we have assumed Vg = 0 without loss of generality. From (7) and (8) theu-component can be written asd2udt2= −f2(u− Ug) .(9)The solution to this equation isu− Ug = A sin(ft) +B cos(ft) ,(10)where the parameters A and B can be determined from the initial conditions. It can beseen from (10) that the solution for the wind oscillates about the geostrophic value. Theperiod of oscillations is 2π/f , which gives for midlatitudes a value of about 17 hours.This type of oscillations commonly occurs in stable atmospheric boundary layer duringthe night. During the day the winds in the boundary layer are below the geostrophic valuebecause of strong turbulence and drag at the surface. When night comes the turbulencedecreases considerably and the pressure gradient accelerates the wind. However, as seenin equations from (7) to (10), the Coriolis term induces an oscillation.In the flow from land to sea a similar situation must occur. Therefore we should expectthe following behaviour in the surface boundary layer: apart from a daily variation onland associated to the radiation cycle, entering the sea the surface wind speed will showan obvious increase due to lower drag on the sea with respect to land. However, atthe same time we expect also the appearance of long oscillations superimposed to theaverage trend, whose frequency depends on the latitude, and the amplitude depends on178 L. CAVALERI and J. TEIXEIRATable I. – Parameters used for the simulation on land and on the sea. The momentum roughnesslength on the sea is wind dependent.Parameter Land SeaVegetation cover 0.867 0.0Momentum roughness length 0.485 0.001Heat roughness length (log.) −4.072 −9.210Background albedo 0.168 0.05Background LW emissivity 0.996 0.996the air-sea stability conditions, For more details on inertial oscillations in geophysicalflows see, e.g., Dutton [18].3. – The organisation of the testsFor the basic test the model was set up for middle latitudes (41 degrees north) andthe month of June, therefore for summer conditions and a marked daily cycle. Thegeostrophic wind was set up at Ug = −1.7 m/s, Vg = −10.0 m/s, with 31 levels in thevertical.Two situations were modelled, on land and on the sea. The respective parameters aregiven in table I. Note that the momentum roughness length on the sea is given by theCharnock relationship z0 = α · u2∗/g. The corresponding figure in the table representsonly the value of the first step.Starting from the initial conditions, the system was left to evolve, i.e. to be advected,on land following the daily cycle. The time step was 15 minutes. The transition (advec-tion) to the sea was simulated shifting to the use of sea parameters, starting from thelatest conditions reached on land.4. – Main resultsThe basic result is shown in fig. 1, showing the evolution in time of the two componentsand the absolute value of the 10 metre wind U10. The time scale is given as integrationsteps, 15 minutes each, hence every small tick corresponds to three hours, while the larger(numbered) ones are one day each. The left part of the figure shows the evolution onland (4 days, beginning and ending at midnight). After 384 integration steps we shift tosea conditions. Note that, because the column of air is advected by the general motion,its evolution in time represents also its evolution in space.The oscillations on land represent the daily cycle, with an intensification of the longi-tudinal v-component during the day, due to sun radiation, and the appearance of unstablesurface conditions.The situation changes dramatically at the transition to the sea. There is an expectedsudden increase of both the wind components, that is associated with the drastic decreaseof the surface friction. This appears as an overshoot of about 0.5 m/s, followed by asequence of regular oscillations, whose period corresponds to the inertial one, about 17hours. After the initial marked overshoot, there is a mild tendency for a decreasingamplitude of the oscillations with time passing.Expectably, locating the tests at different latitudes changes the characteristics of theoscillations. Their frequency decreases moving the test area towards the equator, inOSCILLATIONS IN AN OFFSHORE BLOWING WIND 179Fig. 1. – Time evolution of the surface wind speed U10 on land, followed by a transition to sea(at the vertical line). Continuous line = modulus, broken lines = wind components. Horizontalunits are integration time steps, 15 minutes each. The location is 41 degree north, during June.correspondence to the lower value of the Coriolis term, and increases for the oppositereason towards the pole. The corresponding results for latitudes 21 and 71 degrees northare shown respectively in fig. 2a and b. Changing the latitude leaves the amplitude ofthe oscillations almost unaffected. There is an expected decrease (increase) of the meanvalue of U10 and the peak of the overshoot moving towards the equator (the pole), dueto the decrease (increase) of f , which leads to values of u and v smaller and further from(larger and closer to) the geostrophic wind values.The tests can be, and have been, changed, varying all the basic characteristics of therun, but for the present purposes we show here only one extreme case, that illustrates thedynamics of the system. Figure 3 shows the output of the test at the equator. Note thedifferent time scale, now 2+2 days. The lack of any inertial forcing leads to the completeabsence of overshoot and oscillations, with a simple tendency to a lower equilibrium valuefor U10 than in the other cases, reached in about 12 hours, with a value similar to thatpresent on land during the day.The initial overshoot at the transition to sea is part of the transient during whichthe wind adapts gradually to the new surface boundary conditions. As regards thequestion that stimulated the tests, i.e. if the transient could justify the underestimationof the model results in the Mediterranean Sea, and in general in the enclosed basins (see,e.g., [19, 2b]), we feel the reply is negative. Even if the following oscillations impede anaccurate evaluation of the duration of the transient, as seen in fig. 2 and fig. 3, this seemstoo short to justify the extended underestimation by the model.180 L. CAVALERI and J. TEIXEIRAFig. 2. – a) As fig. 1, but at 21 degrees north; b) as fig. 1, but at 71 degrees north.OSCILLATIONS IN AN OFFSHORE BLOWING WIND 181Fig. 3. – As fig. 1, but at the equator. Note also the different time scale, now 2+2 days.5. – DiscussionOur results, based on the numerical integration of the 1C model, indicate that, whenpassing from land to sea, the wind fields experience some oscillations with an amplitudeof the order of ±10% of the mean U10 value. The oscillations, still depending on theinitial conditions, are strictly related to the inertial forcing of the system. Thereforetheir frequency depends on the latitude. At intermediate latitudes the sudden decreaseof surface roughness from land to sea leads to an increase of the 10 metre wind speed,with a marked initial overshoot.When the test is performed at 0 degree latitude (the period is June), passing to the seathe wind shows a simple asymptotic approach to a much lower value than at intermediatelatitudes.As regards the original question on the response time of the surface boundary layerwhen passing from land to sea, according to our results this is of the order of a few hours.Also because of the presence of an overshoot, the transient boundary layer does not seemto be the complete explanation for the underestimation of the wind speed, hence of thewave height, in the case of an offshore blowing wind.The question arises if the above results depend on the specific choice done whenorganising the tests. The values used for the geostrophic wind and the land roughnesslength were obtained from the 3D ECMWF weather forecast model, for the particularsituation we were interested in, the transition from land to sea in the midlatitudes. Weused different values for the geostrophic wind and the roughness length, in order to test182 L. CAVALERI and J. TEIXEIRAthe sensitivity of the one-column model experiments to these parameters. These differentvalues for the roughness length and geostrophic wind had no impact on the overallconclusions of the study, as expected from the theoretical analysis from subsect. 2.2.It is also quite clear that the oscillations reported in this paper are not of numericalnature. On the one hand, we have performed the same one-column model experimentswith different time-steps (7.5 and 30 minutes), and no substantial difference was notice-able on the results. On the other hand, the period of the oscillations and its latitudinaldistribution coincide with what is theoretical expected (see subsect. 2.2).We have searched the literature for evidence, in favour of or against, our findings, butwe could not find any experiment or related results dealing with the prescribed problem.Therefore we offer it as a hypothesis to be verified or rejected by suitable experimentaldata. On the other hand, pure inertial oscillations like the ones described in this paperare probably also very difficult to find in nature. This is because other forces, likefriction, with different dominant space and time scales, are also controlling the evolutionof momentum, and therefore interacting with, and masking, the inertial oscillations.In this respect it is worthwhile to point out at least one approximation in the waywe framed the experiment. We have assumed that the roughness of the sea surface isdirectly related to the wind speed via the Charnock relationship. There are indications(see, e.g., [1], p. 248) that in the early stages of development of the sea, like the onesexisting off the coast with an offshore blowing wind, the roughness increases substantiallywith respect to the value it approaches at a more mature stage. If this is the case, alarger initial roughness would decrease the value of the surface wind speed just off thecoast, and possibly smooth the overshoot. A proper treatment of this aspect wouldhave required a coupling between the 1C and a wave model, an effort that, given theapproximations involved, was beyond our present purposes.∗ ∗ ∗The numerical experiments have been performed during a staying of one of the authors(LC) at ECMWF. The experiments have been suggested by Anthony Hollingsworth.REFERENCES[1] Komen G. J., Cavaleri L., Donelan M., Hasselmann K., Hasselmann S. andJanssen P. A. E. M., Dynamics and Modelling of Ocean Waves (Cambridge UniversityPress) 1994.[2] Mørk G. and Barstow S., a) Comparison of WAM estimates of Hs withTOPEX/POSEIDON altimeter data in the Mediterranean Sea; a comparison of low andhigh resolution wind models, Oceanor Report OCN R-98019, Trondheim, Norway (1998);b) Comparison of WAM estimates of Hs with TOPEX/POSEIDON altimeter data inEuropean Atlantic waters; a comparison of low and high resolution wind models, OceanorReport OCN R-98025, Trondheim, Norway (1998).[3] Teixeira J., The One-Column Model – Reference and User’s Guide, ECMWF, Reading,UK (1997).[4] Simmons A., ECMWF Newsletter, 56 (1991) 3.[5] Duynkerke P. G., Jonker P. J., Chlond A., van Zanten M.C., Cuxart J., ClarkP., Sanchez E., Martin G., Lenderink G. and Teixeira J., Boundary-Layer Meteorol.,92 (1999) 453.[6] Teixeira J., Q. J. R. Meteorol. Soc., 125 (1999) 529.[7] Bretherton C. S., Krueger S. K., Wyant M. C., Bechtold P., van MeijgaardE., Stevens B. and Teixeira J., Boundary-Layer Meteorol., 93 (1999) 341.OSCILLATIONS IN AN OFFSHORE BLOWING WIND 183[8] Simmons A. and Burridge D., Mon. Weather Rev., 109 (1981) 758.[9] Morcrette J. J., Mon. Weather Rev., 118 (1990) 847.[10] Tiedtke M., Mon. Weather Rev., 121 (1993) 3040.[11] Tiedtke M., Mon. Weather Rev., 117 (1989) 1779.[12] Beljaars A. C. M. and Betts A. K., Validation of the boundary layer representationin the ECMWF model, in Proceedings of the ECMWF Seminar 7-11 September 1992 onValidation of Models over Europe, Vol. II, 1993, pp. 159-196.[13] Louis J. F., Tiedtke M. and Geleyn J. F., A short history of the operational PBLparameterization at ECMWF, Workshop on Boundary Layer Parameterization, November1981, ECMWF, Reading, UK (1982).[14] Beljaars A. C. M. and Holtslag A. A. M., J. Appl. Meteorol., 30 (1991) 327.[15] Lott F. and Miller M. J., Q. J. R. Meteorol. Soc., 123 (1997) 101.[16] Viterbo P. and Beljaars A. C. M., J. Climate, 8 (1995) 2716.[17] Viterbo P., Beljaars A. C. M., Mahfouf J.-F. and Teixeira J., Q. J. R. Meteorol.Soc., 125 (1999) 2401.[18] Dutton J., The Ceaseless Wind. An Introduction to the Theory of Atmospheric Motion(Dover Publications, New York) 1986.[19] Cavaleri L. and Bertotti L., Mon. Weather Rev., 125 (1997) 1964.

Oscillations in an offshore blowing wind

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Oscillations in an offshore blowing wind

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