
    John C. RANSMEIER, Administrator of the Estate of Louis Neil Mariani, Deceased, Plaintiff-Appellee, and Colgan Air Inc., A Virginia Corporation, U.S. Airways, Inc., A Delaware Corporation, L 3 Communications Corporation Security and Detection Systems, A Delaware Corporation, L 3 Communications Corporation, A Delaware Corporation, L 3 Communications Holdings, Inc., A Delaware Corporation, Invision Technologies, Inc., State of Incorporation Unknown, Quantum Magnetics, Inc., State Of Incorporation Unknown, Heimann Systems Corp., State OF Incorporation Unknown, Air France, A French Corporation, Delta Airlines, A Corporation, Swiss, A Swiss Corporation, Air Jamaica, A Jamaican Corporation, Cape Air, Air Transport Association, A Trade Organization, Defendants, UAL Corporation, An Illinois Corporation, United Airlines, Inc., An Illinois Corporation, Huntleigh USA Corporation, A Missouri Corporation, ICTS International NV, A Netherlands Business Entity of Unknown Form, Global Aviation Services, A Delaware Corporation, Burns International Security Services Corp., A Delaware Corporation, Securitas AB, A Swedish Business Entity Of Unknown Form, Massachusetts Port Authority, A Government Entity, The Boeing Company, An Illinois Corporation, Midwest Express Airlines, Inc., A Wisconsin Corporation, Continental Airlines, Inc., A Corporation, Does, 1 Through 100, Inclusive, Midwest Airlines, Inc., Defendants-Appellees, v. Ellen MARIANI, Proposed Intervenor, Appellant.
    
    Nos. 11-175-cv(L), 11-640-cv(Con).
    United States Court of Appeals, Second Circuit.
    June 26, 2012.
    
      Bruce Leichty, Clovis, CA, for Appellant.
    Peter G. Beeson (Charles R. Capace, Zimble & Brettler, Boston, Massachusetts, on the brief), Devine Millimet & Branch, Professional Association, Concord, NY, for Plaintiff-Appellee.
    Jeffrey J. Ellis (Michael R. Feagley, Mayer Brown, LLP, Chicago, Illinois, on the brief), Quirk and Bakalor, P.C., New York, NY, for Defendants-Appellees.
    PRESENT: PETER W. HALL, and SUSAN L. CARNEY, Circuit Judges.
    
      
      . The Clerk of the Court is requested to amend the caption as set forth above.
    
    
      
      . Hon. Barrington D. Parker, Jr., originally assigned to this panel, recused himself from consideration of this appeal. The appeal was decided by the panel’s remaining two judges, who are in agreement as to the disposition, pursuant to Internal Operating Procedure E(b), formerly § 0.14(b) of the Local Rules of the United States Court of Appeals for the Second Circuit.
    
   SUMMARY ORDER

Appellant Ellen Mariani appeals from the district court’s November 15, 2010, order denying her motion to intervene and the March 1, 2010, order denying (once again) her attorney’s motion for admission pro hac vice. We assume the parties’ familiarity with the facts, procedural history, and issues on appeal.

A thorough examination of the record demonstrates that Mariani has only one true argument on appeal — that she was entitled to intervene as of right under Fed.R.Civ.P. 24(a)(2). Rule 24(a)(2), of course, was the very basis for Mariani’s first motion to intervene. The district court denied that motion, and we affirmed that decision on appeal. See N.S. Windows, LLC v. Minoru Yamasaki Associates, Inc., 351 Fed.Appx. 461, 467 (2d Cir.2009) (summary order). Mariani’s renewed attempt to intervene is foreclosed by the doctrine of the law of the case.

This doctrine, in reality, consists of two closely-related rules. The first, the so-called mandate rule, “requires a trial court to follow an appellate court’s previous ruling on an issue in the same case.” United States v. Quintieri, 306 F.3d 1217, 1225 (2d Cir.2002) (emphasis added). Under the second rule, which is somewhat more flexible, a court — be it a district court or an appeals court — will generally follow its own earlier ruling on an issue in later stages of a litigation unless “cogent and compelling reasons militate otherwise.” Id. (internal quotation marks omitted).

Mariani argues that her second motion to intervene does not implicate the law of the case doctrine because it presented a brand-new reason for intervening, to wit, her alleged discovery that Ransmeier was operating under a conflict of interest. The flaw in that assertion, however, is that the district court did not rely on this new argument when it determined to deny Mariani’s second motion to intervene. Rather, the court denied the motion for the same reason as the first time around— that Mariani did not have an interest in the litigation.

The first time Mariani tried to intervene, her chief argument was that a so-called “individual loss of consortium” claim constituted a sufficient “interest” to give her the absolute right, under Rule 24(a)(2), to intervene in the proceedings below. The district court definitively rejected that argument in its November 5, 2007, order denying intervention, holding that, by virtue of her agreement with Peters, Mariani had “no legal status” in the federal action and that any arguments she had regarding her individual loss of consortium claim needed to be made in New Hampshire probate court. As the district court explained, federal courts “do[] not sit to review decisions of the probate court.”

On appeal, this court expanded on Judge Hellerstein’s analysis. Between the time the district court denied Mariani’s motion and the time we decided her first appeal, the New Hampshire Supreme Court had raised some doubt regarding whether Mariani even had an individual loss of consortium claim, and, if so, whether Ran-smeier had the authority to settle that claim. See N.S. Windows, 351 Fed.Appx. at 466-67. We held, however, that Maria-ni’s probate court agreement with Peters demonstrated her clear intention and commitment to abandon all her claims, including her loss of consortium claims, and to let Ransmeier pursue them on her behalf in the Peters litigation. Id. Like Judge Hellerstein, we explained that for the purposes of the federal litigation Mariani was bound by her agreement in New Hampshire probate court. Id. at 467. If she wanted to challenge that agreement, she had to do so in the probate court. Id.

The necessary implication of our decision in N.S. Windows is that, by handing over her claims to Ransmeier, Mariani no longer possessed a sufficient interest to justify intervention as of right under Fed. R.Civ.P. 24(a)(2). To gain such an interest, she had somehow to dissolve the agreement she had reached in the probate court, which is why we pointed her back there. Consequently, in deciding Maria-ni’s second motion to intervene, Judge Hellerstein was not just permitted, he was required to follow our prior ruling. See DeWeerth v. Baldinger, 38 F.3d 1266, 1271 (2d Cir.1994) (law of the case doctrine applies to issues previously decided by necessary implication). And that is exactly what he did, denying the motion on the basis of his own prior ruling and our decision in N.S. Windows. Crucially, although Mariani had briefed her new argument regarding Ransmeier’s alleged conflict in great detail, Judge Hellerstein did not at all rely on this new argument, which arguably goes to the “adequate representation” prong of Rule 24(a)(2). Instead, he explained that he had “already ruled that Ms. Mariani has no interest to justify intervention because her complaints belong before the New Hampshire Probate Court,” and that, despite Mariani’s new arguments, there was “no basis to reconsider” his prior decision. (Emphasis added.)

Mariani may have a right to have a court hear her concerns regarding Ran-smeier’s representation. For example, to the extent Ransmeier’s alleged conflict of interest compromised his obligations to her as administrator of her husband’s estate, she may be able to pursue those claims in New Hampshire probate court. We express no view in that regard. We are firm in our holding, however, that such claims simply do not give rise to an “interest” sufficient to give her the right to intervene in these proceedings.

The district court’s decision to apply correctly the law of the case doctrine and to deny Mariani’s second motion to intervene was not an abuse of discretion. See United States v. City of New York, 198 F.3d 360, 364 (2d Cir.1999). Having properly determined for the second time that Mariani was not permitted to intervene, the district court again did not abuse its discretion in denying Mariani’s attorney’s application for admission pro hac vice on the grounds that Mariani was a non-party.

We have considered the remainder of Mariani’s arguments and find them to be without merit, and therefore AFFIRM the judgment of the district court.

Further, “we cannot help but register our concern with appellant[’s] frivolous conduct in pursuing this appeal.” Smith v. Silverman (In re Smith), 645 F.3d 186, 190 (2d Cir.2011). Despite the clear res judicata effect of N.S. Windows, Mariani and her attorney, Bruce Leichty, chose to return to the district court and file a series of vexatious motions whose lack of legal merit is matched only by their discreditable tone. Beyond specious arguments, including that N.S. Windows had actually given Mariani standing, contra 351 Fed. Appx. at 464, their briefs feature an escalating series of ad hominem attacks on opposing counsel and bombastic challenges to the integrity of the district court. All this has continued on appeal, culminating with Mariani’s and Leichty’s defiant motion to supplement the record, which supposedly identifies “newly discovered” evidence of the district court’s partiality, but is in fact nothing more than a vehicle for asserting deeply troubling personal slurs against Judge Hellerstein and his family.

Under Federal Rule of Appellate Procedure 38, 28 U.S.C. § 1927, and our own inherent authority “to consider sanctions on parties who pursue patently frivolous appeals” and force us to consider and the appellees to defend vexatious litigation, “we may, with adequate notice and opportunity to be heard, impose sanctions nostra sponte.” Gallop v. Cheney, 642 F.3d 364, 370 (2d Cir.2011) (alterations and punctuation omitted). Appellant Mariani and her counsel Leichty are therefore each ordered to show cause, no later than fourteen days following entry of this order, why they should not be sanctioned in the form of double costs, for which Mariani and Leichty would be jointly and severally liable. Their responses shall each be no more than ten pages, double-spaced. Any appellee wishing to respond may file a supplemental letter brief, not longer than fifteen pages, double-spaced, within fourteen days of the date by which the responses of appellant Mariani and her counsel Leichty are due. 
      
      .Mariani also purports to appeal from the district court's January 26, 2011, compromise order authorizing the final distribution of the settlement, and from the district court's February 8, 2011, order denying her motion to reconsider the January order. Because we affirm the district court’s order denying Mar-iani leave to intervene, she has no standing as a non-party to challenge the settlement.
     
      
      . Mariani also asserted at one time that she was a "real party in interest” as defined by Fed.R.Civ.P. 17(a), but she abandoned that position long ago.
     
      
      . Because Judge Hellerstein denied Leichty’s application to appear pro hac vice, Mariani’s briefing below was allegedly filed pro se. We agree with the district court’s assessment that the briefs were obviously written by Leichty. In any event, Leichty's behavior has continued during this appeal, now that he officially represents Mariani.
     
      
      . This motion will be denied by separate order.
     