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Coupled oceanic and atmospheric Ekman layers in the tropics adapted to the central pacific ocean

Gekoppelte ozeanische und atmosphärische Ekman-Schichten in den Tropen in Anpassung an den zentralen Pazifischen Ozean

Couplage des couches d'Ekman de l'Océan et de l'Atmosphère sous les Tropiques; example adapté au Pacifique Central

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Summary

An idealizing analytic solution of the latitudinally differentiated equation of horizontal motion is applied. It neglects temporal and zonal variations and it supposes the zonally averaged distributions of pressure to be prescribed. The solution couples both the Ekman layers by the surface stress being assumed proportionally to the square of the difference of velocities of sea and air at the Prandtl layer. Thermal effects of the air-sea interaction are ignored.

The resulting winds and currents of a numerical example adapted to the Pacific Ocean allow for a continuous passage to mid-latitudes and between the hemispheres, and they show essential properties of the scheme of surface currents. Moreover a “South Equatorial Counter Current” in a southern zone of doldrums can be induced without the company of a westward ascent of the sea surface but in the case of lacking westward wind stress and physically substantiated latitudinal variation of the sea surface level. The model yields the occuring “Southern Doldrums” (or even weak westerly winds) in that near equatorial zone where the northward gradient of surface air pressure reduces to its value at the equator (or falls short of it, respectively).

Zusammenfassung

Zonal gemittelte Verteilungen des Druckes als gegeben voraussetzend wird eine idealisierende analytische Lösung der nach der Meridionalkoordinate abgeleiteten Bewegungsgleichung angewandt, welche zonale und zeitliche Änderungen vernachlässigt, jedoch beide Ekman-Schichten durch den Windschub an der Meeresoberfläche proportional dem Quadrat der Differenzgeschwindigkeit von Luft und Wasser an der Prandtl-Schicht koppelt. Thermische Wechselwirkung bleibt unbeachtet.

Die in einem numerischen Beispiel für den Pazifischen Ozean resultierenden meridionalen und vertikalen Geschwindigkeitsverteilungen weisen einen kontinuierlichen Übergang zu denen mittlerer Breiten sowie von der Nord- zur Südhalbkugel auf und liefern wesentliche Züge des beobachtbaren Systems der äquatornahen Oberflächenströme einschließlich eines “Südäquatorialen Gegenstromes” in einer südlichen Kalmenzone. Bei entsprechender Meridionalverteilung der Wasserspiegellage, für die es physikalische Argumente gibt, bedarf es dazu keines Wasserspiegelanstiegs nach Westen. Die auslösenden Kalmen (ggfs. sogar leichte Westwinde) entstehen nach dem Modell in äquatornahen Breiten des SO-Passats dort, wo der nordwärtige Gradient des bodennahen Luftdrucks auf seinen äquatorialen Wert zurückgeht, bzw. darunter sinkt.

Résumé

Une solution analytique optimisée de l'équation du mouvement horizontal en latitude est mise en application. On néglige les variations dans le temps et dans la zone et on suppose que les distributions moyennes par zone des pressions sont connues. La solution assure le couplage des couches d'Ekman par le frottement tangentiel, celui-ci étant pris proportionnel au carré de la différence des vitesses de l'eau et de l'air dans la couche limite de Prandtl. Les effets thermiques de l'interaction Océan-Atmosphère sont ignorés.

Les vents et courants issus d'un exemple numérique adapté au Pacifique tiennent compte d'une transition continue vers les latitudes moyennes et entre les hemisphéres et montrent les caractéristiques essentielles du régime des courants de surface. D'ailleurs un contre courant Sud Equatorial dans la partie Sud des calmes équatoriaux peut être induit sans être associé à une élévation à l'ouest de la surface de la mer, mais dans le cas d'une absence de friction du vent dans la direction ouest et d'une variation physique importante en latitude du niveau de la mer. Le modèle restitue des zones de calmes au sud (ou même de faibles vents d'ouest) en cette zone de proche équateure où le gradient de pression de l'air en surface en direction du nord se réduit à sa valeur à l'Equateur (ou presque).

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Abbreviations

a 0 =a 0 (y) :

\(\sqrt {f/2K}\) used in eq. (12)

b 0 =b 0 (y) :

\(\sqrt {f/2\aleph }\) used in eq. (13)

C d :

drag coefficient corresponding to the windv d atz = d, C d = 1.3·10−3

D = D(y) :

frictional depth in the ocean whereV f can be neglected (Fig. 1)

d :

base levelz = d, q = 0 of the atmospheric Ekman layer (Fig. 1) If used as index,d indicates the value of variables atq=0 in the atmosphere where the vertical coordinatesz andq = z−d>0

F = F(y) :

factor in the partial solutionv b in eq. (16)

f=2ωsinϕ :

Coriolis parameter, ϕ latitude, ω angular velocity of the earth

g :

acceleration due to gravity

H = H(y) :

frictional height of the atmosphere wherev f can be neglected

h = h(y) :

level of an isobaric surface, e. g. atp=1010 hPa

\(i = \sqrt { - 1}\) :

for representation of the solution by means of complex numbers

i, j, k :

vectors of unity in the directionsx→E,y→N,z→zenith

K :

vertical eddy viscosity in the air;K=4.5 m2/s

m :

defined in (20),m 0=0.01834

0:

if used above a symbol it indicates the value of the respective variable at the equatory=0 or ϕ=0, e. g.\(\mathop v\limits^ \circ _d , \mathop V\limits^ \circ _s , \mathop v\limits^ \circ _G , \mathop \delta \limits^ \circ _d\)

p = p(y,z) :

static pressure in the air and the sea

q||z :

vertical coordinate defined in (5), used in the integration of (9), (10);q<0 in the ocean,q>0 in the atmosphere

r :

radius of the earth corresponding to the surface levelz=0 at ϕ=±4°

R=R(|v d-V s|)S=S(|v d-V s|):

defined in (18)

s :

surface levelz=s, q=0 of the oceanic Ekman layer (Fig. 1),s<0; if used as index,s indicates the value of variables atq=0 in the ocean where the vertical coordinatesz andq=z−s<0

u, U :

zonal component of the velocity in the air and the sea

ν,V :

meridional component of the velocity in the air and the sea

v h,V h :

velocity vectors in the air and the sea, horizontal

v f,V f v b,V b :

parts ofv h, andV h respectively

v G,V G :

tropical β-plane velocities in the air and the sea

u G,U G :

zonal components ofv G andV G respectively

x, y, z :

cartesian coordinates,x→E,y=r·ϕ→N,z→zenith

α:

=α0 (1-iɛ) in eq. (12), α0=0.798·10−3 m−1=α0 (ϕ=±2°), ɛ=1.51

β:

=β0 (1-iχ) in eq. (13), β0=0.01128 m−1=b 0 (ϕ=±2°), χ=0.5226

δd δs δτ :

\(\left. \begin{gathered} \sphericalangle (v_d ,i) \hfill \\ \sphericalangle (V_s ,i) \hfill \\ \sphericalangle (\tau _s ,i) \hfill \\ \end{gathered} \right\}\) see Fig. 11

ζ=ζ(y):

sea surface level, ζ=0 at ϕ=±4°; if used as index, ζ indicates the value of variables immediately at the surface, e. g.v ϕ, τϕ

ϰ:

vertical eddy viscosity in the ocean; ϰ=0.02 m2/s

ϱ:

density of the air; ϱ=1.25 kg/m3

σ:

density of the water; σ=1022 kg/m3

τh :

shearing stress vector in the horizontal plane

τs :

surface stress vector; τs=τζ=τs because ∂τh/∂τz=0 within the air-sea Prandtl layers<z<d (Fig. 1)

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  1. Hans Peter Schmitz

    Present address: , Marienstraße 7c, 1000, Berlin 45

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    Schmitz, H.P. Coupled oceanic and atmospheric Ekman layers in the tropics adapted to the central pacific ocean. Deutsche Hydrographische Zeitschrift 40, 157–179 (1987). https://doi.org/10.1007/BF02282617

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