Observations and Modelling of Fronts and Frontogenesis

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follows from (2a) and (8b) that the potential vorticity in layer 2, P2 (f u2y)/h2 (12a) obeys the equation, 2t + V2P2y wP2/h2. (12b) Similarly, when layer 3 is in contact with the mixed layer, so that (9b) holds instead of (8c), P3 obeys, instead of (ha). 3t + V3P3y weP3/h3, (13) Vertical integration of the hydrostatic equations yields, in the three-layer domain, P1 p0(y,t) gzp1(y,t), (l4a) P2 = p0(y,t) g[p - p1(y,t)]h1(y,t) (14b) p3 = p0(y,t) gp3 - p1(y,t)]h1(y,t) g(p3 - p2)h2(y,t) - gpz, (14c) where p(y,t) is surface pressure. In the two-layer domain, (14c) holds with h2 0, and (14b) is neglected. Inserting (14) into the geostrophic relations (la) and (2a) yields the thermal wind relations, p0f(u1 u2) g[(p Pl)hly + (l/2)h1(p2 Pl)y] ' (iSa) p0f(u2 - u3) = g(p3 - p2)(h1 + h2)y or, in the two-layer domain, = g(p3 - 2)h3, (15b) p0f(u1 - u3) g[(p3 - pi)hi + (l/2)h1(p3 Pi)yIl (15c) 54
Equations (la) and (2a) are dynamic equations for the geostrophic momentum in each layer. Equation (3) is a thermodynamic equation for the mixed layer density. Equations (8) or (9) are kinematic equations for the layer depths. These evolution equations all depend on the ageostrophic velocities vj. However, there is no evolution equation for the v, since their time derivatives have been eliminated by the semigeostrophic approximation. Since the geostrophic shear (15) depends only on the layer depths and densities, the rate of change of the geostrophic shear may be calculated in two different ways: either directly, from the momentum equations (la) and (2a), or indirectly, from the kinematic and thermodynamic equations (3) and (8) or (9) for the layer depths and mixed layer density. This calculation yields consistency relations that the ageostrophic velocities v must satisfy at each time in lieu of the missing evolution equations. These relations take the form of differential equations in y. In the three-layer domain, taking a time derivative of (l5a,b) and substituting (la), (2a), (3), (6), and (8) yields, g(p0f)[(p2 Pl)hl2hllyly h1(f u2y)(Vl - v2) TX/p0 + g(2p0fY-{h1[czQ/c + (P2 Pl)we])y, (l6a) g(pof)(p3 - p2)(h3v3) + (f u2y)v2 - (f u3y)v3 = 0. (16b) 55
Page 1 and 2:
AN ABSTRACT OF THE THESIS OF Roger
Page 3 and 4:
Observations and Modelling of Front
Page 5 and 6:
To my parents, Hans and Nancy Samel
Page 7 and 8:
This research was supported by ONR
Page 9 and 10:
TABLE OF CONTENTS I. INTRODUCTION 1
Page 11 and 12:
Figure LIST OF FIGURES (continued)
Page 13 and 14:
OBSERVATIONS AND MODELLING OF FRONT
Page 15 and 16: II. TOWED THERMISTOR CHAIN OBSERVAT
Page 17 and 18: oriented multiple surface fronts up
Page 19 and 20: gravity waves. A maximum of 21 and
Page 21 and 22: small, since the large-scale temper
Page 23 and 24: front at about 18.5 °C. There is f
Page 25 and 26: given in the stable cases (downward
Page 27 and 28: temperatures for January and Februa
Page 29 and 30: dominates the spectrum. At these wa
Page 31 and 32: typically iO-2iO3 °c m1-), and sin
Page 33 and 34: 11.5 Geostrophic turbulence We have
Page 35 and 36: energy of each of the longitudinal
Page 37 and 38: varying N and a flow isolated from
Page 39 and 40: adiabatic meridional displacements
Page 41 and 42: z0 34 32 V v 30 a -J 28 26 a) - 2b$
Page 43 and 44: U0 V L 0 L V 20 E 16 V 12 p ; t *,
Page 45 and 46: 30 E 50 -c a- I, 0 70 90 300 360 37
Page 47 and 48: 10 10° 10-2 1 0 -3.0 -2.0 -1.0 0.0
Page 49 and 50: 1 o 1 102 101 - 0 LU 00 x I.- - -1
Page 51 and 52: E (I) a. z 106 1 o5 102 101 100 1O
Page 53 and 54: Table 11.1 Tow start and finish pos
Page 55 and 56: Table 11.3 Climatological sea surfa
Page 57 and 58: Paulson, C. A., R. J. Baumann, L. M
Page 59 and 60: 111.1 Introduction In this chapter
Page 61 and 62: interior layers, on the horizontal
Page 63 and 64: polynomials in z.) The vertical mom
Page 65: hit + (hivi) = We, (8a) h2t + (h2v2
Page 69 and 70: subdivided into subdomains with dif
Page 71 and 72: mixed layer. As is the case with (2
Page 73 and 74: constant pressure c = 4.1x103 J K1-
Page 75 and 76: When (28) have been solved for the
Page 77 and 78: occur with the initial conditions w
Page 79 and 80: analytically. Let Al2 and A23 be th
Page 81 and 82: Shortly after t = 8.1, the mixed la
Page 83 and 84: that around the offshore front, exc
Page 85 and 86: the reduced equations numerically.
Page 87 and 88: (1) The upwelled horizontal profile
Page 89 and 90: Figure 111.1 Model geometry. zO z z
Page 91 and 92: * j 0. -20. >\ - -40. 0 > .0 0. 40.
Page 93 and 94: 4) 20. 10. 0. 20. 1 0. 0. 0 y/X 0.
Page 95 and 96: BIBLIOGRAPHY Batchelor, G. K. , 195
Page 97 and 98: Rhines, P. B., 1979: Geostrophic Tu
Page 99 and 100: APPENDIX A: DYNAMICAL DERIVATION OF
Page 101 and 102: In a manner exactly analogous to th
Page 103 and 104: smoothness. The large vorticities t
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