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Characterizing Transport Between the Surface Mixed Layer and the Ocean Interior with a Forward and Adjoint Global Ocean Transport Model

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Characterizing Transport Between the Surface Mixed Layer and the Ocean Interior with a Forward and Adjoint Global Ocean Transport Model APRIL 2005 P R I M E A U 545 Characterizing Transport between the Surface Mixed Layer and the Ocean Interior with a Forward and Adjoint Global Ocean Transport Model FRANÇOIS PRIMEAU Department of Earth System Science, University of California, Irvine, Irvine, California (Manuscript received 1 July 2003, in final form 7 September 2004) ABSTRACT The theory of first-passage time distribution functions and its extension to last-passage time distribution functions are applied to the problem of tracking the movement of water masses to and from the surface mixed layer in a global ocean general circulation model. The first-passage time distribution function is used to determine in a probabilistic sense when and where a fluid element will make its first contact with the surface as a function of its position in the ocean interior. The last-passage time distribution is used to determine when and where a fluid element made its last contact with the surface. A computationally efficient method is presented for recursively computing the first few moments of the first- and last-passage time distributions by directly inverting the forward and adjoint transport operator. This approach allows integrated transport information to be obtained directly from the differential form of the transport operator without the need to perform lengthy multitracer time integration of the transport equations. The method, which relies on the stationarity of the transport operator, is applied to the time-averaged transport operator obtained from a three-dimensional global ocean simulation performed with an OGCM. With this approach, the author (i) computes surface maps showing the fraction of the total ocean volume per unit area that ventilates at each point on the surface of the ocean, (ii) partitions interior water masses based on their formation region at the surface, and (iii) computes the three-dimensional spatial distribution of the mean and standard deviation of the age distribution of water. 1. Introduction between successive visits to the surface mixed layer. Since fluid element properties are reset by air–sea in- The interaction of the ocean with the rest of the cli- teractions during each visit to the surface mixed layer, mate system happens primarily at the sea surface the “memory” of a fluid element does not precede its through air–sea fluxes. These air–sea fluxes imprint the last contact with the surface layer, nor does it persist current physical and chemical state of the atmosphere beyond its next contact with the surface layer. Ocean on water parcels. When water parcels are transported transport is therefore described by the distribution of below the surface, they become shielded from the at- times for fluid elements to have had their last contact mosphere until they resurface and communicate past with the atmosphere as well as the distribution of times physical and chemical climate conditions to the atmo- when fluid elements will have their first (or next) con- sphere. The delay between successive visits to the tact with the atmosphere. mixed layer imparts to the earth system an important The analysis of geophysical flows in terms of distri- long-term memory. Understanding the duration of this bution of times for fluid elements to be transported memory is more complicated than the simple “con- from a surface of interest to the interior of the fluid veyor belt” picture of the global ocean circulation, body is a growing subject. A useful review in the con- which suggests that fluid parcels are transported below text of stratospheric air can be found in Waugh and the surface at one end of the conveyor belt, carried steadily along, and fatefully popped up at the other end. Hall (2002). In an oceanographic context, Haine and In reality fluid elements are transported to and from Hall (2002) combined the familiar concepts of water the mixed layer by advective, mixing, and diffusive pro- mass and age into a generalized theory formulated in cesses that lead to a continuous distribution of times terms of the surface boundary condition propagator (boundary Green function) of the ocean’s advection– diffusion transport equation. The connection between the forward and adjoint transport equations and the Corresponding author address: Dr. François Primeau, Depart- ment of Earth System Science, Croul Hall, University of Califor- distribution of times for fluid elements to be trans- nia, Irvine, Irvine, CA 92602. ported to and from a surface is discussed in Holzer and E-mail: [email protected] Hall (2000). Delhez and Deleersnijder (2002), formu- © 2005 American Meteorological Society JPO2699 546 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 late a more general theory that allows the age of a ement in the ocean interior at time t, its residence time tracer constituent to be tracked. The connection be- can be decomposed into two parts, tween the surface-to-interior transit time distribution ␶ ϭ ␶ ϩ ␶ ͑ ͒ and the different concepts of tracer age used by ocean- r lp fp, 1 ographers is discussed in Hall and Haine (2002) and in ␶ where the first part, lp, is the time that has elapsed Waugh et al. (2003). since the fluid element made its last contact with the sea In this paper we explore when, where, and how much ␶ surface and the second part, fp, is the time that will fluid is exchanged between the surface mixed layer and pass before the fluid element makes its next contact the interior ocean in a global ocean general circulation ␶ with the surface. We refer to fp as the first-passage model (OGCM) by computing first- and last-passage time in accordance with the terminology used in the 1 The quantities we compute are not time distributions. literature on stochastic processes (e.g., Stratonovich only useful diagnostics for characterizing the ventila- ␶ 1963). By analogy, we will refer to lp as the last-passage tion of an OGCM but are properties of the ocean that time. The last-passage time is also refered to as the are not directly observable but that are nevertheless of transit-time or age in the oceanographic literature (e.g., direct interest to oceanographers studying the exchange Haine and Hall 2002; Wunsch 2002). between the ocean and the atmosphere. The of CO2 Because of diffusive and mixing processes, the fluid focus of this paper is not on the transport of any par- elements in a parcel are transported to and from the ticular tracer, but rather on the physical transport of surface via many pathways, and a distribution is needed fluid elements caused by currents and mixing processes. to describe when and where the fluid elements in a The plan of the paper is as follows. In section 2 we water parcel make their last and first contact with the introduce the first- and last-passage time distributions sea surface. and their connection to the residence time of a fluid To make these ideas precise, we will denote the parcel in the ocean interior. The transport problems ocean surface by ⍀ , a point on the surface by x , and a that must be solved to compute the distributions and s s point in the ocean interior by x (Fig. 1). The joint prob- their moments are also given in this section. In section ability density that a fluid element was at x on ⍀ at 3 we describe the offline ocean transport model and the s s time t Ϫ ␶ , given it is currently at interior position x at dynamical OGCM. Section 5 presents surface maps lp time t, is denoted by L(x , ␶ |x, t). The joint pdf that a that show the volume fraction of the total ocean volume s lp fluid element at position x at time t will make its first that ventilates at each surface point of the ocean. Sec- ϩ ␶ contact with the surface at time t fp and and position tion 6 presents a watermass analysis for the model that ␶ | xs, will be denoted by F(xs, fp x, t). was computed using the last-passage time distribution. In the following, we will focus on the situation where Section 7 presents an age analysis as well as an analysis the circulation is assumed to be steady. In other words, of the time for water in the interior to return to the we will assume that the transport by time-dependent ocean surface. Section 7a discusses the shape of typical eddies can be represented by a stationary eddy diffu- last-passage time distributions. Section 8 discusses the sion tensor. For the stationary case, the joint distribu- results and concludes. ␶ ␶ tion for (xs, lp) and (xs, fp) are independent of t and are thus conditioned only on the interior position x. 2. First- and last-passage time distributions a. Last-passage time problem formulation for a We begin by considering a fluid element located be- stationary flow low the sea surface in the ocean interior, and denote by ␶ r , the total amount of time the fluid element will spend As shown in Holzer and Hall (2000) and in Haine ␶ | in the ocean interior—that is, the amount of time be- and Hall (2002), the joint distribution, L(xs, lp x), for ␶ tween its last contact with the sea surface and its next the last-passage time, lp, and last surface-contact posi- contact with the sea surface. If we consider a fluid el- tion, xs, can be expressed in terms of the propagator of Ϫ | ϭ Ј surface boundary conditions L(xs, t ts x) G (x, t; xs, Ј ts), where G is the solution to 1 In the literature on the theory of stochastic processes, transit- time distributions are known as first-passage time distributions Ѩ GЈ͑x, t; x , t ͒ ϩ TGЈ͑x, t; x , t ͒ ϭ 0, ͑2͒ (Stratonovich 1963).
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